Time-dependent variational Monte Carlo

The time-dependent variational Monte Carlo (t-VMC) method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

${\displaystyle \Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)}$

where the complex-valued ${\displaystyle a_{k}(t)}$ are time-dependent variational parameters, ${\displaystyle X}$ denotes a many-body configuration and ${\displaystyle O_{k}(X)}$ are time-independent operators that define the specific ansatz. The time evolution of the parameters ${\displaystyle a_{k}(t)}$ can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion

${\displaystyle i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},}$

where ${\displaystyle {\mathcal {H}}}$ is the Hamiltonian of the system, ${\displaystyle \langle AB\rangle _{t}^{c}=\langle AB\rangle _{t}-\langle A\rangle _{t}\langle B\rangle _{t}}$ are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., ${\displaystyle \langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle }$.

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret ${\displaystyle {\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{2}\,dX}}}$ as a probability distribution function over the multi-dimensional space spanned by the many-body configurations ${\displaystyle X}$. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time ${\displaystyle t}$, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories ${\displaystyle a(t)}$ of the variational parameters are then found upon numerical integration of the associated differential equation.